The former Bachelorette welcomed her second child, son Austin Michael, with husband Stephen Stagliano last month, and it's been a blissful time. "We're really happy," she tells PEOPLE. "We're just tired. The longest stretch I get is four hours. But it's a good tired!"

Pappas Stagliano and her husband, Stephen, were committed to having their two children have the same initials. Austin was the "only A name [Stephen] really liked for a boy," she says, and Michael is for Stephen's brother, Bachelorette alum Michael Stagliano, who introduced the couple.

The growing pains to a family of four have been few and far between – big sister Addison took to her little brother immediately. "Addison is obsessed with him, which I'm very happy about," Pappas Stagliano says. "She loves him, she wants to kiss him all the time. She says, 'I love you baby Austin.' Before she goes to bed at night she comes in to give him a kiss and a hug. It’s been a pretty flawless transition."

Like many other new moms, Pappas Stagliano went to Pinterest to find her nursery inspiration – and decided to do a wood wall. "All they had to do was stain it," she says. "It has a rustic look. I love it. They did the wall in less than a day!" The finished product includes his Windsor crib from the new furniture brand évolur and lots of Mabels Labels, which were used to personalize his baby gear.

Pappas Stagliano was also committed to keeping Austin's room very "minimalist," but it was important to her to display his books in his "READ" nook. "I didn't really want a lot going on in his room," she says.